Proper Holomorphic Mappings in the Special Class of Reinhardt Domains

A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in C with the logarithmic image equal to a strip or a half-plane is given. 1. Statement of results We adopt here the standard notations from complex analysis. Given γ = (γ1, γ2) ∈ R 2 and z = (z1, z2) ∈ C 2 for which it makes sense we put |z | = |z1| γ1 |z2| γ2 . The unit disc in C is denoted by D and the set of proper holomorphic mappings between domains D,G ⊂ C is denoted by Prop(D,G). In this paper we deal with the pseudoconvex Reinhardt domains in C whose logarithmic image is equal to a strip or a half-plane. Observe that such domains are always algebraically equivalent to domains of the form Dα,r−,r+ := {z ∈ C 2 : r < |z| < r}, where α = (α1, α2) ∈ (R )∗, 0 < r + < ∞, −∞ < r < r. We say that Dα,r−,r+ is of the irrational type if α1/α2 ∈ R \ Q. In the other case Dα,r−,r+ is said to be of the rational type. Recall that if r < 0 < r, α ∈ (R)∗, then the domains Dα,r−,r+ are so-called elementary Reinhardt domains. Below we shall give a complete description of all proper holomorphic mappings between the domains Dα,r− 1 ,r + 1 and Dβ,r− 2 ,r + 2 for arbitrary α, β ∈ (R)∗ and 0 < r + i < ∞, −∞ < r − i < r + i , i = 1, 2. Similar problems were studied in some papers. In [Shi1] and [Shi2] the problem of holomorphic equivalence of elementary Reinhardt domains was considered. These results were partially extended by A. Edigarian and W. Zwonek. In the paper [Edi-Zwo] the authors gave a characterization of proper holomorphic mappings between elementary Reinhardt domains of the rational type. 1991 Mathematics Subject Classification. 32H35; 32A07.